A Mathematical Gift, 1: The Interplay Between Topology, Functions, Geometry, and Algebra (Mathematical World) (v. 1) by Kenji Ueno (PDF)

Description

This book will bring the beauty and fun of mathematics to the classroom. It offers serious mathematics in a lively, reader-friendly style. Included are exercises and many figures illustrating the main concepts.

The first chapter presents the geometry and topology of surfaces. Among other topics, the authors discuss the Poincaré-Hopf theorem on critical points of vector fields on surfaces and the Gauss-Bonnet theorem on the relation between curvature and topology (the Euler characteristic). The second chapter addresses various aspects of the concept of dimension, including the Peano curve and the Poincaré approach. Also addressed is the structure of three-dimensional manifolds. In particular, it is proved that the three-dimensional sphere is the union of two doughnuts.

This is the first of three volumes originating from a series of lectures given by the authors at Kyoto University (Japan).

User’s Reviews

Editorial Reviews: Review “With A Mathematical Gift, I, there is no reason why every undergraduate student should not be exposed to some topology … accessible to even high school students … beautiful illustrations and straightforward explanations of sophisticated ideas. Real world and concrete scenarios are used … elegant explanations … exercises … are friendly and non-threatening … the perfect choice for anyone who conducts summer workshops for high school students … an ideal supplement for graduate students studying topology for the first time … also excellent for undergraduate independent studies … I enjoyed reading this book … fun to look at … instructive and motivating … impressed … wowed by the detail and clarity presented by the authors. Readers of A Mathematical Gift, I will want to read A Mathematical Gift, II.” —- MAA Online

Reviews from Amazon users which were colected at the time this book was published on the website:

⭐I’m a masters student who recently completed an introductory course in differential geometry, and another in algebraic topology. I enjoyed the courses and did well, but if I’d read this book beforehand, the going would’ve been even better.The authors’ love of the subject and love of teaching shines through. The book is written so that even advanced high school students can follow it, and yet it’s deep enough to spark the imagination of this grad student and remind me of the magic of manifolds and space.Provides excellent and intuitive overviews and proofs on the Euler characteristic, Poincare-Hopf theorem, Gauss-Bonnet, different notions of dimension, higher dimensional manifolds, and various related topics.When I teach geometry or topology, I will definitely include “A Mathematical Gift” in the syllabus.I look forward to checking out volumes II and III.

⭐Simply put, this is a great book.It is definitely not a ‘serious’ math book (no definitions, no proofs) but it is very well written and provides a nice, intuitive introduction to many advanced topics.Should be required reading for math majors at the freshman or sophomore level!

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